equation 3, of § 7, becomes 
and, therefore, 
AND VIBRATION OF SHAFTS. 
303 
d'^yldx'^ — () .( 1 ), 
y = ^ ^ Ox D .(2), 
If, as before (§ 8), undashed symbols refer to the symbols or constants on the left, 
and dashed symbols to those on the right of a singular point, then (as in Chapter II., 
equations 7-16) we shall have precisely the same differential equations holding at 
the specified singular points, the only difference being that when those differential 
equations are integrated, the forms of the resulting equations are altered from a 
trigonometrical (in Chapter II.) to an algebraic form in the present case. 
22. It is now proposed to investigate some of the cases, commonly occurring in 
practice, according to the second method of solution. Whatever be the manner 
in which the shaft is supported, the effect of the shaft is neglected, and the shaft 
supposed to be loaded with one pulley only. 
The effect of the shaft, and of more than one pulley, will be considered in §§ 59-62. 
Case IX. 
23. Overhanging shaft, length c, fixed in direction at one end, and 
LOADED WITH A PULLEY, WEIGHT W, AND MOMENT OF INERTIA, 1' AT ITS END. 
Thus— 
Fig. 12. 
We have (§ 21, equation 2), 
y = ^x^-\- + Ox 4- D. 
0 
Taking the origin at the shoulders, we have, when x = 0, 
y == 0, dyjdx = 0, 
whence 
D = 0.(1), 
C = 0..(2). 
